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Knowledge base artificial intelligence.pdf
1. 1
International Islamic University Chittagong
January 22, 2024
Tashin Hossain
Department of Computer Science and Engineering
Lecture#7
Agents that Reason Logically
And Propositional Logic
Adjunct Lecturer
2. Course Information
2
Course Materials:
1. Stuart Russell, Peter Norvig: “Artificial Intelligence A
Modern Approach”, 3rd Edition.
2. Other specific materials and lecture slides
3. Course Information
3
Segment Contents
03
Game playing: Introduction, Perfect Decisions, Imperfect
Decisions, Alpha-Beta Pruning.
Constraints satisfaction problem: Travelling salesman problem,
graph coloring etc.
04
Propositional and First-Order logic: Knowledge
Representation, Reasoning and Logic; Propositional Logic:
Syntax, Semantics, Validity and Inference, Rules of Inference for
Propositional logic; First-Order Logic: Syntax and Semantics,
Using first-order logic.
Inference in first order logic: Inference Rules Involving
Quantifiers, Example Proof, Generalized Modus Ponens, Forward
and Backward Chaining, Completeness, Resolution.
4. Course Information
4
Segment Contents
05
Probabilistic Reasoning: Probability and Bayes’ Theorem,
Certainty Factors and Rule-Based Systems, Bayesian Networks,
Fuzzy Logic; Ds theory, ER
Some Expert Systems: Representation and Using Domain
Knowledge, Expert System Shells, Explanation, And Knowledge
Acquisition.
06
Learning: Introduction to Learning, Inductive Learning, Learning
Decision Trees, Neural Net Learning;
07
Natural language processing: Introduction, Syntactic
Processing, Semantic Analysis, Discourse and Pragmatic
Processing.
6. Knowledge and Reasoning
6
From the previous lectures it has been shown that an agent that
has goals and searches for solutions to the goals can do better
than one that just reacts to its environment (simple reflex agent)
We focused mainly on the question of how to carry out the search
(using different search strategies), leaving aside the question of
general methods for describing states and actions
7. Knowledge and Reasoning
7
It is necessary to powering the agents with the capacity for general
logical reasoning
A logical, knowledge-based agent begins with some knowledge of the
world and of its own actions
It uses logical reasoning to maintain a description of the world as new
precepts arrive and to deduce a course of action that will achieve its goals
In this lecture we introduce the basic design for a knowledge-
based agent, followed by a simple logical language for expressing
knowledge, and show how it can be used to draw conclusions
about the world and to decide what to do
8. A Knowledge Based Agent
8
A knowledge-based agent consists of a knowledge base
(KB) and an inference engine (IE)
A knowledge-base is a set of representations of what one
knows about the world (objects and classes of objects, the
fact about objects, relationships among objects, etc.)
Each individual representation is called a sentence
The sentences are expressed in a knowledge representation
language
9. A Knowledge Based Agent
9
Examples of sentences
– The moon is made of green cheese Fact
– If A is true then B is true Rule
– A is false
– All humans are mortal
– Confucius is a human
The Inference engine derives new sentences from the input
and KB
The inference mechanism depends on representation in KB
10. A Knowledge Based Agent
10
The agent operates as follows:
1. It receives percepts from environment
2. It computes what action it should perform (by IE and KB)
3. It performs the chosen action (some actions are simply
inserting inferred new facts into KB).
12. KB can be Viewed at Different Levels
12
Knowledge Level
– The most abstract level
– describe agent by saying what it knows
– Example: A taxi agent might know that the Golden Gate Bridge
connects San Francisco with the Marin County
Logical Level
– The level at which the knowledge is encoded into sentences
– Example: Links(GoldenGateBridge, SanFrancisco, Marin-County)
Implementation Level
– The physical representation of the sentences in the logical level
– Example: “Links (GoldenGateBridge, SanFrancisco, MarinCounty)”
14. What is Proposition?
14
A proposition is a statement that can be either true or
false; it must be one or the other, and it cannot be both
The sun rises in the east
Today is a very hot day
16. Logic is a Formal Language
16
Propositional Logic
Rahima is intelligent - proposition
Rahima is hardworking – proposition
If Rahima is intelligent and Rahima is hardworking
Then Rahima scores high marks -proposition
17. Elements of PL
17
Rahima is intelligent => intelligent ( Rahima)
Rahima is hardworking
Objects and Relations or functions
Elements
18. Towards the Syntax
18
Intelligent (Rahima) == Rahima is intelligent
Hardworking (Rahima) == Rahima is hardworking
19. Towards more syntax
19
o Let P stand for intelligent (Rahima)
o Let Q stand for Hardworking (Rahima) F
o What does P ˄ Q (P and Q) mean ?
o What does P ˅ Q (P and Q) mean ?
o P ˄ Q, P ˅ Q are compound propositions
20. Syntactic Elements of PL
20
o A set of propositional symbols (P, Q, R etc.)
o each of which can be True or False
o Set of logical operators :˄, ˅, Not, implies
o often parenthesis() is used for grouping
o There are two special symbols
o TRUE (T) and FALSE (F) – these are logical constants
21. How to form propositional sentence?
21
– Each symbol (a proposition or a constant) is a sentence
– If P is a sentence and Q is a sentence, Then
– (p) is a sentence
– P ˄ Q is a sentence
– P or Q is a sentence
– not P is a sentence
– P implies Q is a sentence
– Nothing else is a sentence
– Sentences are called world form formulae (wff)
22. Examples of WFFs
22
P , Q True
– P and Q True
– (P or Q) implies R If the roads get wet, then it rains
– (P and Q) or R implies S
– Not (P or Q)
– Not (P or Q) implies R and S
23. Implication
23
P implies Q P Q
If P is true then Q is true
If it rains then the roads are wet
Sufficient condition
If the roads are wet then it rains ???
24. Equivalence
24
P bidirectional Q
Triangle
If two sides (A,B) of an triangle are equal (P), then
Opposite angles are equal (Q)
If opposite angles of a triangle are equal (Q), then two
sides of an triangle are equal (P)
25. What does a wff mean -Semantics
25
Interpretation in a world
When we interpret a sentence in a world we assign
meaning to it and it evaluates to either True or False
The Child can write
Nursery
Class 2
F
T
World
Semantics
26. Semantic of compound sentence
26
P: Likes ( Rahima, Sanjida)
Q: Knows (Tania, Farzana)
World: Rahima and Sanjida are friends and
Tania and Farzana known to each other
P= T, Q=T
P and Q =T
P and Not Q = F
27. Validity of a sentence
27
If a propositional sentence is true under all possible
interpretation, it is VALID
• Tautology
• Satisfiability
• Inconsistent
P v ¬ P = T
Question:
28. Validity of a sentence
28
• Tautology Compound Proposition is always True
• Satisfiability A compound sentence is satisfiable if there is at least
one true result in its Truth table
• Valid: A compound proposition is valid when it is a tautology
• Invalid: A compound proposition is invalid when it is either a
contradiction or contingency
29. Validity of a sentence
29
Inconsistency in propositional logic refers to a situation
where a set of propositions or statements leads to a
contradiction, meaning that it's impossible for all the
statements in the set to be simultaneously true
Let's consider a simple example:
Statement A: It is raining.
Statement B: It is not raining.
30. Quiz
30
Express the following English sentence in the
language of PL
– It rains in July rains ( July)
– The book is not costly not costly (book)
– If it rains today and one does not carry umbrella, then
he will be drenched
Rains(today) ^ not carry( umbrella) drenched(he)
31. Quiz
31
If P is true and Q is true then are the followings true or
false
– P → Q
– (¬P∨Q) → Q
– (¬P^Q) → P
– P ∨ ¬P
33. Objectives
33
• Infer the truth value of a proposition
• Reason towards new facts, given a set of propositions
• Prove a proposition given a set of propositional facts
34. Procedure to derive Truth Value
34
P Q P^Q PvQ
F F F F
F T F T
T F F T
T T T T
P v Q will be True when any of them is True
P ^ Q will be True when both of them are True
35. Procedure to derive Truth Value
35
P Q P Q
F F T
F T T
T F F
T T T
If you won’t reach the station on time, then you won’t be able to catch
the train T
If P is True, then Q is True, since P is a sufficient condition so that Q is
True
36. De Morgan’s Law
36
1. ¬ ( P ^ Q ) = ¬P v ¬ Q
2. ¬ ( P v Q ) = ¬P ^ ¬ Q
Prove the De morgan’s law using Truth
table
37. Reasoning
37
• P: It is the month of July
• Q: It rains
• R: P→Q
• If it is the month of July then it rains
• It is the month of July
• Conclude: It rains
39. Reasoning –Modus Ponens
39
• Modus Ponens is very much a valid inference
rule prove
P Q ¬P v Q
Prove:
(¬P v Q) ^ P
(¬P v P) v Q
T v Q
Q
40. Valid inference rule
40
• Thus Irrespective of meaning modus ponens allow us to
infer the truth of Q
• Modus Ponens is a inference rule that allow us to
deduce the truth of a consequent depending on the truth
of an antecedents
51. Converting a compound proposition to
the clausal form
51
Example: Consider the Sentence
¬ ( A B) v (C A)
1. Eliminate the Implication Sign
¬ (¬A v B) v (¬ C v A)
2. Eliminate the double negation and reduce scope of “not”
signs
(A ^ ¬ B) v (¬C v A)
3. Convert to conjunctive normal form by using distributive and
associative law
(A ^ ¬ B) v (¬C v A) => Distributive Law: A(B+C) = AB + AC
( A v ¬C v A) ^ (¬B v ¬C v A)
( A v ¬C) ^ (¬B v ¬C v A)
4. Get the set of clauses
( A v ¬C)
(¬B v ¬C v A)
